Nonlinear spherical caps and associated plate and membrane problems

This paper deals with the existence and multiplicity problem of the equilib rium solutions of an elastic spherical cap within nonlinear strain theory. We pose the problem in the form of a three parameter bifurcation problem, o ne parameter being related to the load, the others to the geometry. When th e geometrical parameters are different from zero, the so-called generic cas e, we revisit the nonuniqueness results, and explore the solutions in the p arameter space. Then we analyze the formal limits as the geometrical parame ters tend to zero. When the curvature tends to zero, we obtain from the non linear shell a von Karman plate, a new, although natural, result. When the thickness parameter tends to zero, we get a nonlinear membrane problem. A s tudy of the latter gives infinitely many solutions, and we discuss the cons truction, shapes, and stability in detail.